Monday, 25 August 2014

Interlude: A Bunch Of Monsters!

Well in the last week or so a whole bunch of monsters have made their way to us! There was an update to Horde Of The Dragon Queen, the Player's Handbook was released, a new version of Basic D&D was released with over 150 monsters and there have been a number of Monster Manual creature spoilers released.

As a result my sample size jumped from 33 monsters to 209!

On top of that there have been several tweaks to PC data via the PC Stats thread on the Wizards of the Coast forums.

All this means one thing... It's time to stop and review what we have done so far. This will delay the next scheduled post a few days, but it's important to do.

I think it's best to cover this all in one big post, so forgive me if this one gets a bit long. To make it up to you here's a handy-dandy index...

Challenge Rating

Most of the mechanical items we talked about in the CR section of Part 3 have either been ratified or remain unchanged. The DM Basic Rules v0.1 released at Basic D&D does provide some insight and a couple of useful tables that replace some of what we discussed.

The Proficiency Bonus By Challenge Rating table on page 4 shows that we were correct about using CR to select a monster's proficiency. It also handily extends monster Proficieny below CR1 and above CR20 for us.

The Experience Points By Challenge Rating table on page 5 shows that we were very close with our XP values, at least in the CR0 through CR10 range. If we take our formula and extend out past CR10 to CR30 we find some inaccuracies creep in, though. As I previously said, our small set of sample data made it unwise to project out past CR10.

Rather than smply work off a formula for the curve Wizards of the Coast have, in effect, spliced three curves togethor. What I can see by CR is this...

CR <1: These creatures look like they are hand-fashioned off of CR1 creatures. One could probably model the first and second curves togethor by going to a fairly complex polynomial.

CR 21+: Another curve greatly increases XP gain for defeating monsters, which is appropriate since PCs will supposedly be fighting creatures far more powerful than themselves. This appears to be a polynomial equation similar to the following:
XP = 564.39x^2 - 15148x + 101214

This is pretty much of academic interest, but it does serve to illustrate the dangers of computing data values based on too small a sample size.

Reflection: I predicted that we would see creatures of CR30 or even beyond. The Tarrasque spoiler from the Monster Manual is CR30, confirming my prediction.

Size

As with Challenge Rating, most of the elements we discussed in the Size section of Part 3 are ratified or remain unchanged by the DM Basic Rules v0.1 in the recent Basic D&D release.

The Hit Dice By Size on page 3 of the DM Basic Rules v0.1 shows that our belief regarding the link between Size and Hit Dice was correct and that we estimated the Hit Dice size for Huge and Gargantuan creatures correctly.

Naturally there were no references to the linkage between Size and Speed.

Reflection: It's a bit hard to draw any conclusions about my prediction about monster sizes at higher levels, since we still have very few higher-CR monsters to look at. But at this stage I see no reason to doubt my prediction.

Ability Scores

On reviewing Ability Score values and referring back to the Ability Score section of Part 4 I find that the greater sample data has upheld the findings of that post. In fact the trendlines we use have firmed up very considerably and now naturally align quite closely to the formulae and table we constructed in there! I can only see this as statistical ratification.

I'd be quite comfortable extending this out to CR15 based on the data we now have.

Reflection: While there is no definative proof of the prediction I made that peak attributes would reach 30 past CR20 I feel very comfortable clarifying that this will become the common peak Ability Score closer to CR30 than CR25.

Hit Points

Before considering the impact the new mass of monsters has on the Hit Points section of Part 4, let's stop a moment and consider the community input on my assessment of PC statistics. It's important to consider this first because, as we said in part 4, these aspects of monster math are built on PC math.

Over the last couple of weeks the Wizards of the Coast community has discussed this on my PC Stats thread over on the WotC forums. A number of changes have been suggested to the way I have calculated this data and most of it has been incorporated into my math. But the funny things is, at the end of the day it hasn't significantly changed the curves I described in part 4. Yes, some of the variations in the progression changed. But in the end the curve formula remained unchanged. Which makes things a bit easier in this review.

Now the new monsters... Gaining such a significant injection new data showed up something interesting things about the relationship between PC Damage and monster Hit Points! I found the new data pretty interesting and spent a fair bit of time re-analyzing HP as a result.

It turns out the PC Damage:Monster HP ratio isn't a static 5, as previously posted. It's very clear that the multiplier actually starts somewhere down near 3.5 or 4 and trends up very quickly to an average of 7 then scales out to around 12 at CR30.

I did some pretty extensive analysis of this, attacking the problem from several different angles. By the end I had several potential formulae that could potentially be used to build monster HP. But profiling these I found most of my methods of correlation pointed towards the use of a Logarithmic expression to determine the multipiler. I believe it's close to the following...
Multiplier=0.31 x LN(CR) + 3.7
Which yields a monster HP formula something like this...
HP=(4.55 x CR ^ 0.72) * (0.31 x LN(CR) + 3.7)

It's a little convoluted, but testing showed some positive flags...

A close relationship to observable HP data on both scatter and summary graphs.

It's unlikely any single formula can adequately match all sampled data - which is a significant factor behind my belief that CR assessment is a measure of composite data. However this formula achieves a high proportion of match, with its trendline being mainly centred within sample data.

And the result of this relates very well to a simple linear equation which can be used in its place seamlessly...
HP=20 x CR + 8

Now that I had reset the monster HP progression values I had to re-baseline the monster HD table. This is a job that takes a couple of hours or so by hand and I had already done it three times. So I invested a couple of hours writing up a VBA macro to compute it all. Now with the press of a button it's all recalculated in less than a second.

Being an ex-programmer pays off from time to time!

Reflection: I indicated that I believed monster Hit Points would be more variable than monster Damage, once we were able to analyse the Monster Manual. I've seen nothing to suggest that I should recant this.

Armor Class

The significant increase in sample size has served to clarify AC significantly. In fact sample size is now big enough for us to realistically estimate AC independently of PC Attack Bonus, if we choose to do so. This opens up several avenues of correlation that we did not have in the Armor Class section of Part 5.

Examining trendlines on scatter and summary graphs reveals a very linear progression of monster AC, with values still fairly tightly grouped at +/-5 of trendlines. Note that when looking past CR20 we do need to be careful of skew as both the CR24 Ancient Red Dragon and the CR30 Tarrasque likely have an AC near the top of their range.

AC curve vs linear

Reassessing our data from a PC Attack Bonus perspective also shows that the relationship between it and AC has clarified. Between CR1 and CR20 we can see that same-level PCs consistently need to roll an 8 or better to hit the monster. Considering that PC Attack Bonus is a curve how do we resolve this to a linear monster AC progression?

Well, it's not as difficult as some readers might expect.

Simply plotting the two on the same graph provides some understanding of this. The red line on the graph shows PC Attack Bonus plus 8 (in other words, it's a pure derivation). The blue line represents a linear equation closely matching the summary data I have crunched in my spreadsheet. You'll se that the results of the two will be quite close, particularly after rounding.

CR

Armor Class

Average

0

13

⅛

13

¼

13

½

13

1

13

2

13

3

13

4

14

5

14

6

14

7

15

8

15

9

15

10

16

Now some readers might feel concerned about the small disparities between the two. But as we previously noted Bounded Accuracy means that AC varies more within a given CR than it does across all CRs. And it's very normal to see AC variance up to +/-5 of normal within a given CR's monsters. We even see variation beyond this occasionally. So it really is something we simply do not need to be concerned with.

Readers will note that several resulting AC values are different on the new table, mainly at the lowest levels. But the differences are pretty minimal and this again highlights the reasons behind our decision to restrict ourselves to CR10 and below until the Monster Manual is released.

Reflection: It looks like average monster AC may peak at 22 at CR29 or CR30, rather than at CR25.

Miscellaneous Stats

While the scatter graph contained many more data points its trendline remained completely unchanged. The summary graph also gained more data points (because we have sample monsters at levels where we previously did not) and this changed the high-value end of the trendline... Only a small amount.

Reflection: I've not seen anything to suggest that I should change the predictions I made in this area.

Example: Human Pyromancer

Let's see how our findings impact our sample monster. We'll also make a couple of other minor tweaks/corrections to the Pyromancer...

Human Pyromancer

Medium humanoid (human), any alignment

Armor Class 12

Hit Points88 (16d8+16)

Speed 30ft

STR9 (-1)

DEX14 (+2)

CON12 (+1)

INT19 (+4)

WIS12 (+1)

CHA12 (+1)

Saving Throws Dex +5, Int +7

Skills Arcana +7, Perception +4

Damage Resistances Fire

Lanuguages Common, Ignan

Challenge 5 (1,800 XP)

While our chosen Challenge Rating of 5 remains unchanged we do need to update the Pyromancer's XP reward from 1,700 to 1,800.

Size and Ability Score values remain unchanged.

Based on the updated data the creature's Hit Points are probably a bit too low, even for a glass cannon. We could either increase number of HD or we could increase the Pyromancer's Constitution Score. Since changes in HD generally result in smaller incremental changes I tweaked this up by two to 16, resulting in an average HP of 88.

Although we have made some updates to Armor Class in this post it's important we remember that AC tends to vary more within a given CR than it does when across all CRs, thanks to Bounded Accuracy. As such I believe that AC does not need to be updated for the Pyromancer.

While we don't need to make any changes to Miscellaneous Stats base don this post, I did notice an error in the Pyromancer as listed last installment. I had intended to give it proficiency in Perception, which is much more useful for monsters than Arcana. And I accidentally left off the actual Arcana bonus. I've corrected these in this version and will go back and edit my previous post shortly.

Armor Class

Monster Armor Class has a close relationship with PC Attack Bonus. While our small sample size does present some difficulties here we can do some analysis on PC Attack Bonus to help us leverage value from the samples we have. Yes, there will be some element of uncertainty, but it will be smaller than many people expect.

Monster AC Scatter

If we plot all of the monster AC values by level we get an interesting little scatter graph, with most of the data on the left hand side because of the predominantly low level nature of our sample data. A trendline will be less impacted by this low-level skew, however the sparseness of higher level data does tend to make the trendline somewhat unreliable. But remember, that's why we are constraining ourselves to CR10 and below.

What's important here is that we get a sense of roughly where the average is for the data we have, what we think the trendline should look like and what kind of variability we should expect to see at a given level.

Monster AC Average

We can also average monster AC data and plotting the results on another graph, adding a trendline to this graph too. We can then consider the similarities and differences between the two, along with what these might signify. As you can see this tightens up the clusters and brings the lower level data into sharper focus. Unfortunately this doesn't directly help with our higher level samples.

That said this is probably enough to crack this particular nut for CR0 through CR10... That is, if we do some analysis of related PC data and then leverage off it.

Average PC Attack

To achieve this I had to go and analyze Attack Bonus data for PC Classes, as I had previously done for PC Damage in order to properly analyze monster Hit Points. Since I was doing this anyway I also did the initial analysis on PC AC, updated my previous PC data spreadsheet and floated a new PC Stat Curves thread on the Wizards of the Coast forums. Hopefully this proves useful to others and you are welcome to look at it yourself and comment back on that thread, if you are so inclined.

The casual reader might be forgiven for assuming that we can express the PC Attack Bonus using a simple linear expression, or possibly a power expression. While we can get fairly close to the trendline we see in that graph using a linear or power equation, but to achieve something close to an exact match we need to use a simple polynomial similar to this one:
Attack = -0.0062 x Level ^ 2 + 0.495 x Level + 4.012

This curve is particularly useful for several reasons. It almost perfectly matches the projected progression, having a 0.2 variance that only tells at four points due to rounding - levels 5, 9, 10 and 13. AT these points the rounded value slips one above or below the actual PC class values. I think that falls with the margin for error in the observed PC data itself. The progression below level 1, where we might match with CR, looks appropriate. Past level 20 the progression becomes quite shallow, which should be appropriate for building CR21+ creatures. Additionally the equation is one of the least complicated polynomials. It seems fairly elegant to me.

CR

Armor Class

Average

0

13

⅛

13

¼

13

½

13

1

14

2

14

3

14

4

15

5

15

6

16

7

16

8

17

9

17

10

17

Now, what should an average D&D 5e character roll to hit an average CR-equivalent creature? Subtracting our average PC Attack bonus from the AC of an average CR-equivalent monster we see that it's very close to an average of 9 for our level 1 through 4 sample range. I guess it's possible that it could be an 8 or a 10, but we'll need the kind of data the Monster Manual will supply to work that out for sure. So for now we'll go with a 9.

The table on the left displays these values, rounded to the nearest integer, for CR0 through CR10.

That said it's really important to understand that the there appears to be a margin of at least +/-5 on this average. So it's perfectly legitimate for a CR1 monster to have an AC anywhere from 9 to 19 or for a CR10 monster to have an AC between 12 and 22, without compensating significantly in any of it's other stats.

Furthermore this is a guideline only and experienced 5e monster designers shouldn't have any qualms about going even further outside the boundaries, though they should probably ensure they do apply some kind of mitigation elsewhere in their creature. More inexperienced monster designers should probably aim to stay in or near these boundaries until they gain some confident building creatures with this system.

You see, the D&D 5e monster system "feels" very strongly like a "rule of sense" type system. The emphasis is more on the art of monster design than in D&D 4e, for example. And this is an area where Bounded Accuracy really makes itself felt.

Prediction: Based on my analysis so far I think it's pretty obvious that there is a progression, flattened though it may be, for AC and, by inference, Attack Bonus. On top of that I reckon some folks are going to cry foul when they realise that there is the case. Those people will be missing the point. Bounded Accuracy isn't about there being no progression on this axis. It's about the progression being quite muted so that monsters can stay relevant across more levels.

Prediction: I think average monster AC will peak at about 22 around level 25.

Miscellaneous Stats

In general the miscellaneous statistics (those in the same section as Challenge) seem to follow the rule of common sense. A creature native to the Elemental Plane Of Fire is probably immune to fire damage and an especially sneaky creature in all probability has proficiency in Stealth. So the main driver for miscellaneous stats does appear to be logic.

Additionally it's pretty obvious that D&D 5e monsters strongly leverage off a templating system and this starts to become evident in miscellaneous statistics. Undead are immune to poison damage and both the charmed and poisoned conditions, for example. Since the use of templating layers up attributes on a creature systems that use it usually cautioned not to apply too many templates as the creature may become overpowered for it's level (or CR in the case of D&D 5e). The impact on analysis is to increase the observed variability of creatures, which is consistent with my observations.

With this all said there does appear to be a general progression of these stats as CR increases. Whether this is by design or incidental, simply being a result of increasing complexity and toughness, is a moot point. The pattern is present and, given sufficient data, it can be enumerated.

But do we have enough data? Well yes and no. The release of the Player's Handbook, the update of Horde Of The Dragon Queen and certain spoilers from the Monster Manual have helped and those stats have been included in this part of this post. We can be fairly confident of this pattern out to CR10 and I'll illustrate why shortly.

By experimenting with different ways of combining the miscellaneous stats I came up with a method that keeps the variance minimal and makes sense - simply add up the number of Saving Throws, Skills, Damage Resistances and Damage Immunities the creature has, arriving at a "misc stats score". This method should be considered provisional and how it actually fits with the build process Wizards of the Coast will present remains to be seen. That said it does appear to work with the current crop of monsters.

Monster Misc Stats Score Scatter

Creating a scatter graph of these scores and placing a linear trendline on the graph suggests an interesting pattern. Based on this graph CR1 creatures typically score just over 2 and CR30 creatures score around 11. And we can see at the lower CRs that the variability seems to be about +/-5.

Of course, the dearth of higher level data will generally cause that end of the trendline to be unreliable and to wander. That's why we aren't looking past CR10 until we have more data.

So how can we gain some confidence in the lower level data? Let me illustrate the approach I have used in this particular case.

Monster Misc Stats Score Average

What I did was generate a pivot table of average by CR, scatter graph that and place a linear trendline on the graph. See the Blue data points and blue trendline on the Monster Misc Stats Score Average figure.

As you can see this produces a fairly different trendline to the preceeding graph, primarily on the righthand side, where data is most sparse. This trendline would suggest CR1 creatures score just under 4 and that CR30 creatures scoure about 7.5.

Overlaying the trendline from the first figure in red highlights the disparity

So how do we reconcile these? Splitting the difference is a pretty common-sense approach. Past CR10 things start to become increasingly uncertain. Restricting ourselves to CR10 and below (check out the green box) we can see that the margin for error is very low and that we can have high confidence in this approach at those levels.

Now let's have a look at what I've observed about each of the miscellaneous stats...

Saving Throws

Monsters may add proficiency to their saving throws in the same way as PCs.
Counts towards Misc Stat Score.

Skills

Skills are calculated for monsters in the same way as for PCs. Monsters may add their proficiency modifier to skills they are trained in.
Counts towards Misc Stat Score.

Damage Vulnerabilities

These are pretty rare in D&D 5e, with only 1.8% of current samples having a vulnerability. These don't seem to be used to compensate for other strengthening attributes, though one could use them to provide an Achilles Heal for a creature that is particualrly strong for it's CR.
Best advice here is to only apply a vulnerability where there is an especially compelling reason for your monster to have one.Does not count towards Misc Stat Score.

Damage Resistance

Damage resistance is pretty common in 5e.
Note that resistance to bludgeoning, peircing and slashing from nonmagical/non-adamantine/unsilvered weapons seems to be the replacement for the "immune to damage from nonmagical weapons" in older editions and counts as one Damage Resistance.
Counts towards Misc Stat Score.

Damage Immunities

In D&D 5e these are rarer than Damage Resistances, but 19.4% of the current sample have them.
Given they are somewhat stronger than an equivalent resistance I had hoped weight them as "worth" more than resistances. However, this simply introduced greater skew and variability. I'd suggest that best advice is to go light on Damage Immunities, but to use them where there is a compelling reason in your creature's background.
Counts towards Misc Stat Score.

Condition Immunities

Creatures with resistance or immunity to a particular damage type normally also have immunity to it's corresponding condition, if there is one.
Otherwise these should be applied where there is a compelling background/story reason.Does not count towards Misc Stat Score.

Senses

Most creatures have one additional sense. Some have none, some have two or even more. There doesn't seem to be any consideration here beyond story/background.Does not count towards Misc Stat Score.

Languages

The number of languages a monster knows seems to be purely defined by story/background considerations, or "the rule of sense".Does not count towards Misc Stat Score.

As we can see "the rule of sense" and the concept of "the art of monster building" are alive and well in D&D 5e. And along with those all the baggage and concerns from earlier editions that had the same focus.

CR

Average Misc Stat Score✝

0

1

⅛

1

¼

2

½

2

1

3

2

3

3

3

4

4

5

4

6

4

7

4

8

4

9

5

10

5

✝ Varies by +/-5

So what "score" should we use for each CR? Well using the medium value between the two in our last graph it's pretty easy to come up with a formula that produces appropriate results...
Score = 0.2 x CR + 2.8

Note that I have hand-editted the vlaues for CRs of less than 1 to save complexity. We could produce a solid match using a Power or Poly2 formula, but I don't think this rates the extra time that would be involved.

Prediction: I think Monster Manual analysis will confirm we are on the right track here and that we'll find that CR30 creature do have an average score around 9 using this approach.

Prediction: I expect the Dungeon Master's Guide's section on monster construction will suggest using "the rule of sense".

Example: Human Pyromancer

If we recall our Glass Canon with an Arcane Fire theme we'll remember it isn't going to be especially hard to hit, but focusses on dealing damage.

Human Pyromancer

Medium humanoid (human), any alignment

Armor Class 12

Hit Points 77 (14d8+14)

Speed 30ft

STR9 (-1)

DEX14 (+2)

CON12 (+1)

INT19 (+4)

WIS12 (+1)

CHA12 (+1)

Saving Throws Dex +5, Int +7

Skills Arcana +7, Perception +4

Damage Resistances Fire

Lanuguages Common, Ignan

Challenge 5 (1,700 XP)

That doesn't mean the Pyromancer won't have a couple of little tricks to make it a little more viable. Our plan is for melee characters to get close and then for it to go down quickly.

The Pyronmancer is unarmored so adds it's Dexterity Modifier to it's AC. This puts it on the low side of appropriate for it's CR, which is fine.

As a CR5 creature it'll typically have 4 significant miscellanoues stats, though it may have anywhere from 0 to 9. Now, what is thematic?

As a caster with a casting attribute of Intelligence it makes sense go give it proficiency on that, though proficiency on Dexterity is also appropriate and helps encourage melee characters to deal with it, so I have added both... After it's not that common for monsters to make intelligence saves.

Arcana is an obvious choice for Skills, though of rather limited application for a monster. We'll also give the Pyromancer something useful in combat - proficiency in Perception.

The choice of fire resistance is an obvious one, given the creature's theme.

Given the Pyromancer is human I'll give it Common and an additional language. Ignan, the language of Fire Elementals and their kin, seems like a good fit.

Saturday, 2 August 2014

Where Surf maps out some of the parameters and progression around monster Ability Scores and Hit Points...

Part 4: Construction: Ability Scores & HP

Before getting into things I'd like to thank the guy who forwarded Appendix B of the Horde Of The Dragon Queen PDF to me - you know who you are and you have my thanks. The 5 monsters in it have been entered into my spreadsheets and these all fit analysis to date. The highest CR of these was a CR4 Dragonborn NPC.

Onwards!

D&D 5e monsters rely heavily on their Ability Scores to implement much of their mechanics. Once we have determined these we are in a position to compute dependant stats, like Hit Points. After considering these two aspects of D&D 5e monsters we'll apply the results to our Example Monster.

Ability Scores

I spent some time analysing monster ability scores. I looked at standard arrays. I looked at some point-buy approaches. And I looked at combinations of the two. I also tried to factor in possible racial bonuses. Now it's quite possible a combination of these approaches is being used, but if they are it's something not evident to me.

Raw Max & Average Ability

Analyzed Max & Average Ability

What is apparent is that monster ability scores adhere to an average based approach, at least for the data available at the time of writing.

Dropping a trendline on a scatter graph of each creature's maximum and average ability score produces some interesting results. We can see that the two are generally an equal distance apart. That said, the predominance of the very low level monsters does skew the trendline.

If we do some aggregation on the data we can work out both the average maximum attribute score and the average attribute score for creatures at each CR. We can then create a scatter graph of this data and plot trendlines on the two sets of data. This shows us that average ability score remains pretty consistent and that the relationship between it and average maximum ability score is also normally fairly consistent. There is a little skew due to the Young Green Dragon, but that creature has sacrifices it's high score a little in order to improve some of it's secondary scores.

CR

Ability Scores

Average

Maximum✝

0

8

13

⅛

9

14

¼

10

15

½

10

15

1

11

16

2

11

16

3

12

17

4

12

17

5

13

18

6

13

18

7

14

19

8

14

19

9

15

20

10

15

20

✝ Typical maximum.

This data makes it pretty simple to construct a linear progression out to level 10 that we can be quite confident of. We focus on the CR1 through CR10 creatures for matching the curve and identifying the appropriate formula. For creatures lower than CR1 I simply entered data matching the analysis. I could have built a poly2 or poly3 formula to produce the same results. But I figure people following this series would rather me not waste several hours, possibly delaying the next post by an extra day as a result.

With a few minutes work we can obtain a simple linear formula that fits our requirements very well...
Average Ability=0.5 x CR + 10.5

Our analysis of the data tell us that the maximum attribute value is typically five higher than the average attribute value for the CR, but sometimes they are seven or even more. That's fine as long as the creature's abilities average out to a value close to the average for the Target CR.

Since monster designers will be building these creatures in the context of D&D 5e discarding the remainders is the most appropriate approach here and I have used that method to construct the table for this section.

You can use the top-right table from page 9 of the Basic D&D v0.1 PDF to determine each ability's modifier based on the ability score. If you use spreadsheets the following formula should work perfectly (assuming your data is in cell A1):
=FLOOR((A1-10)/2,1)

Prediction: We know that maximum monster ability scores will be 30. My prediction is that we won't commonly see this in monsters until past CR20. I think it'll plateua shortly after that CR out to the CR25 or CR30 maximum I predicted in my last post.

Hit Points

Monster Hit Points are one of the most important aspects of D&D 5e monsters, along with the creature's damage output. I believe that these two stats are the principle elements used to determine a creature's Actual CR, but we'll look at that in a later post.

During my D&D Next Monster Analysis we saw that there is an important relationship between PC Damage and monster HP. In fact PC Damage Output is the main consideration for determining appropriate monster HP and that is clearly still the case in D&D 5e.

In order to analyse monster Damage and Hit Points I had to enter that same data for PC Classes into a spreadsheet and use that data in the analysis of monster stats. This data was presented to the community for review, comment and reuse in this thread on the Wizards of the Coast forums.

Average/Total PC Damage

Graphing the sum and average of PC Damage reveals some interesting patterns. At levels 5, 11 and 17 PCs see a significant gain in damage output - a jump of about 5 damage on top of their normal progression. In other words they double the damage gain they normally make over an entire tier of play. This kind of "stepping" is often used in game design to help build a feeling that characters make significant gains in power by advancing into new tiers of play. Wizards of the Coast has used this before and even told us this would be the case at these levels in recent Legends & Lore posts and on Twitter.

Wizards of the Coastdidn't tell us about the flatter damage gain in the final tier and it's final damage step at level 20. But this is hardly surprising since PCs typically take on Deities and Greater Demon Lords at this point in the game. Interestingly this also lends some support to the idea that creatures higher than CR20 will be part of the game.

I've read a number of comments on the net about folks finding 5e monsters difficult to figure out. They often point to power curves like those in D&D 4e and talk about how predictable and incremental these are. Folks seem to find the stepping in 5e problematic because they can't look to the underlying curve. The thing is, even if Wizards of the Coast are simply cobbling all of 5e together in a huge spreadsheet without using coherent formulae and curves... Why curves will still tend to be there! And this is no exception.

PC Damage/Monster HP Curve

If we take the average PC damage output from our first graph we can plot a simple Power trendline over it. We can angle this trendline so that much of the regular damage progression in each tier aligns with the trendline. The dip before the step at the end of each tier falls away below this and the steps at levels 5, 11 and 17 (and the one at level 20) pop up to just above the trendline.

And thus the magic of curve-matching reveals to us the actual power curve for PC Damage in D&D 5e...
Damage=4.55 x Level ^ 0.72

So how do the Hit Points of monsters from the Starter Set (and Horde of the Dragon Queen) relate to PC Damage? Well what I did was compare those with the actual PC stats at the same level as the CR in question. For monsters with a CR under CR1 there needed to be a slight manual workaround in the comparison - to gain the base equivalent PC Damage I had to determine it from the level 1 PC data.

CR

Hit Points

Min

Target

Max

0

1

1

3

⅛

3

5

7

¼

6

8

11

½

11

14

19

1

18

23

31

2

29

37

44

3

43

50

57

4

55

62

68

5

67

72

96

6

77

83

78

7

87

92

88

8

97

102

98

9

106

111

107

10

115

119

115

Note: Typical boundaries shown.

To be honest things break down a little bit below CR1 anyway. For example, what's the difference between a CR⅛ and a CR0 creature's hitpoints? We are talking about the difference between say 2hp and 4hp. Most PCs can kill either with a single hit. They differentiate more when we introduce Damage into the equation, but we'll leave that for it's own post.

If we divide the average monster HP by the average PC Damage for the same level/CR we see that it is consistently a 5. What does this mean? Well a party of 4 PCs shaping up against a monster of the same CR as their level can expect that opponent to survive for about two rounds, on average.

From here it's pretty simple to build an appropriate formula for Monster Hit Points. In this case we'll round any remainders to the nearest whole number and assume a minimum value of one. Here's what I came up with...
HP=(4.55 x CR ^ 0.72) x 5

We could assume that the halfway point between each of these values is the cut-off point for each CR, but that isn't the case. There appears to be some overlap between adjacent CRs. This is mainly constrained within a 5% overlap, however the Hit Points of some monsters is significantly above or below these limits. In most of these cases there is a corresponding opposite variation in Damage and the remainder of the cases compensate in some other manner. This supports the supposition that Damage is a significant factor when it comes to assessing Actual CR.

CR

ConMod

Hit Dice

Tiny(d4)

Small(d6)

Medium(d8)

Large(d10)

Huge(d12)

Garg'n(d20)

0

-1

1

-

-

-

-

-

⅛

-1

3

1

1

-

-

-

¼

+0

3

2

2

1

1

-

½

+0

2

3

3

2

2

1

1

+0

9

7

5

3

3

2

2

+0

16

12

9

6

5

4

3

+1

16

12

10

7

6

5

4

+1

21

16

13

10

9

6

5

+1

25

20

16

12

11

8

6

+1

29

23

19

15

14

9

7

+2

26

21

18

15

14

9

8

+2

29

24

20

17

16

11

9

+2

32

27

22

18

17

12

10

+2

35

29

24

20

19

13

Once we have decided on a target Hit Point value for the creature we need to decide how many Hit Dice it should have to be at or near that target. This isn't necessarily as staightforward as one might think and it's best that we understand how Average HP is calculated...
Average HP=(Num_HD x Dice_Avg) + (Num_HD x Con_Mod)

As we can see this pretty much the traditional pre-D&D 4e method for determining monster HP.

The fiddly part here is determining the appropriate HD for a monster based on it's Target HD, incorporating the creature's Constitution modifier into the equation. To help facilitate this I spent a couple of hours precalculating a table of the number of a given HD needed to achieve close to a given Target CR's target Hit Points, assuming the creature has a constitution of average for that Target CR.

Hopefully this helps put folks in the right ballpark when building monsters for a given Target CR. But in many cases readers will find that they still need to tinker a bit with number of HD and the creature's Constitution score. In these cases modifying number of HD generally results in the smallest incremental changes, while changes to Constitution normally result in bigger differences.

So how do the monsters in the Starter Set and in Appendix B of Horde Of The Dragon Queen fit these tables?

Between the Starter Set and Appendix B of Horde Of The Dragon Queen there are 33 creatures.

15 of these 33 have HP outside the boundaries I described.

8 of these 15 could be have inappropriate hitpoints after factoring in their Damage, using my provisional method.

4 of these 8 are more than 3 hitpoints outside the ranges we have discussed. I consider 3 or less to be an acceptable margin of error in this area.

So these 4 creatures could be considered "outside CR", but is this the case?

Flameskull

This creature falls well short of minimum target HP. Looking at the Flameskull we can see that it has considerable resistances and immunities, possibly more than other creatures of this CR will have. But the real trick here seems like the Rejuvenation trait. PCs will probably have to fight this creature twice to beat it! So we can consider it valid. Additionally I did not include Damage from it's Spellcasting trait and this should probably compensate it's HP a modest amount.

Langdedrosa Cyanwrath

This creature from Horde of the Dragon Queen seems to fall a bit short on both HP and Damage. But if we look closely at it we can see that it has quite high Alpha Damage. I suspect that this has been factored in as approximately 7 or more in their base Damage, which would see them within an appropriate Actual CR.

Spectator

While the Spectator's HP are below the minimum for it's CR that's not why it falls below CR. Instead this comes back to difficulties accurately determining the creature's Damage output. Only one of the Spectator's four eye attacks cause damage and so only that eye's damage has been calculated. If we assume that the it's Spell Reflection and the effect of another eye combined would be "worth" at least 12 damage we find that it falls neatly within it's CR.

Zombie

The Zombie's HP are much more than we'd expect for a CR¼ creature, in fact they are more in line with a CR1 monster. On the flip-side it's Damage is more appropriate to a CR⅛ or even a CR0 monster. Combined with it's slow movement I think the Zombie is probably OK. As I've inferred before, the tighter margins below CR1 allow for less leeway with creatures that push the boundaries of our definitions.

Prediction: Once the Monster Manual has been released I believe we'll see that creature Hit Points will vary outside their suggested boundaries more frequently than their Damage does.

Example: Human Pyromancer

As I said in my last post the Pyromance will be a ranged glass cannon with a focus on Arcane fire.

Human Pyromancer

Medium humanoid (human), any alignment

Hit Points: 77 (14d8+14)

Speed: 30ft

STR9 (-1)

DEX14 (+2)

CON12 (+1)

INT19 (+4)

WIS12 (+1)

CHA12 (+1)

Challenge 5 (1,700 XP)

I'm loosely modelling it on the Wizard class, so I will focus on Intelligence as it's main stat. To compensate I will give it below average Strength, because of all those years studying dusty tomes. The Pyromancer's Dexterity will be slightly above average - he looks after his fingers and practices a lot of arcane gestures and exercises. The remaining stats I will leave at average for a CR5 creature.

This monster is going to rely on cover and it's allies to protect it, rather than high defences and an ocean of health. So I am going to set it's Hit Points below the minimum suggested for a CR5 creature. This will allow me to push it's Damage up quite high for it's Actual CR. Tweaking number of HD and not adjusting the creature's Constitution score is the easiest way to do this fine tuning.